optimal systems, and will be examined in the following. First, in (6.20) for the H ¨older’s inequality
M−1
Y
i=0
Z 2π 0
[L−1e ESxxE†L−1e ]iidω 2π ≥
Z 2π 0
M−1
Y
i=0
[L−1e ESxxE†L−1e ]iidω 2π
!M
to have equality, we require the spectrum equalizing property (6.13) to be satisfied. This gives us another necessary condition for the optimal biorthogonal GTD filter banks:
Corollary 6.4.4 Spectrum equalization of Subband Signals is Necessary for Optimal Biorthogonal GTD Filter Banks: The subband signals of the optimal biorthogonal GTD filter banks have the spectrum equalizing
property (6.13). ♦
Second, in (6.21) for the Hadamard inequality
M−1
Y
i=0
[R†R]ii ≥det(R†R)
to have equality,R†Rneeds to be diagonal. This shows that the matrixR(ejω)must be decompos- able as a paraunitary matrix multiplying with a diagonal matrix. Therefore, the optimal solution to (6.7) will definitely have the structure as in Fig. 6.2. This is stated as the following corollary:
Corollary 6.4.5 (Optimal Biorthogonal GTD FBs Necessarily Has a Decomposable Structure) The optimal biorthogonal GTD filter banks have the structure as shown in Fig. 6.2. ♦ To summarize from Theorem 6.4.1, Corollary 6.4.4, and Corollary 6.4.5, the optimal biorthogonal GTD filter bank also hastotal decorrelationandspectrum equalizationas the two necessary conditions.
Furthermore, the optimal systems can be decomposed as the structure in Fig. 6.2. Surprisingly, all of these results have a parallel fashion to the theory of traditional biorthogonal filter banks devel- oped in [109] and [70]: Theorem 2.3 in [70] suggesting that total decorrelation is necessary, Lemma 2.6 in [70] suggesting that spectral majorization is necessary, and Theorem 2.7 in [70] suggesting that the optimal biorthogonal filter banks have the decomposable structure as shown in Fig. 3 of [70].
Table 6.1: Features of Optimal Filter Banks Used in Subband Coders φ(Coding Gain =σx2/φ1/M) Nec. and Suff. Conditions Orth. SBC [106] QM−1
i=0
R2π
0 ηidω2π Total Decor., and Spec. Maj.
Biorth. SBC [70] Ä QM−1
i=0
R2π 0
√ηidω 2π
ä2
Total Decor., Spec. Maj., and opt. orth. FBs + scalar filters Orth. GTD SBC ÄR2π
0 (detSxx)M1 dω2πäM
Total Decor. and Spec. Eq.
Biorth. GTD SBC ÄR2π
0 (detSxx)2M1 dω2πä2M
Total Decor., Spec. Eq., and opt. orth. GTD FBs + scalar filters
mance of the optimal orthonormal GTD SBC and the optimal biorthogonal GTD SBC are exactly thedeterminant bounds(not achievable for most input statistics) derived in [109] (see Eq.(39) and Eq.(38) in [109]) for the orthonormal SBCs and the biorthogonal SBCs, respectively. We also list the necessary and sufficient conditions for the optimal solutions in each case.
In the following we compare the coding gain performance of the optimal subband coders in these four cases. We use an AR(1) test input with parameterρand an AR(2) process with poles at z± = ρe±jθ. The AR(1) process is often used to model simple images in the literature, and AR(2) process models certain types of image texture as mentioned in [69]. The recursion of the autocorrelation function in AR(2) isrn = 2ρcosθrn−1−ρ2rn−2withr0= 1andr1= (2ρcosθ)/(1 + ρ2). We also compare the performance with the theoretical bound on the coding gain, namely the prediction gain given by [107]:Gth=σ2x/exp (Rπ
−πlnSxx(ejω)dω2π),whereSxx(ejω)denotes the psd of signalx(n).
0.854 0.9 0.95
5 6 7 8 9 10 11
ρ
Coding Gain (dB)
Orthonormal Biorthogonal GTD−Orthonormal GTD−Biorthogonal Theoretical Bound
Figure 6.3: Coding gain of subband coders withM = 3for the AR(1) process withρfrom0.85to 0.95.
0.957 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 8
9 10 11 12 13 14 15 16
ρ
Coding Gain (dB)
Orthonormal Biorthogonal GTD−Orthonormal GTD−Biorthogonal Theoretical Bound
Figure 6.4: Coding gain of subband coders withM = 4for the AR(2) process withρfrom0.95to 0.99andθ=π/3.
Fig. 6.3 shows the coding gain forM = 3andρfrom0.85to0.95for the AR(1) input. Fig. 6.4 shows the coding gain forM = 4,ρfrom0.95to0.99andθ=π/3. It can be seen that in both cases the optimal biorthogonal GTD coder is only about0.1dB away from the theoretical bound and is about1dB better than the optimal biorthogonal subband coders [70]. This suggests the advantage of using the GTD filter banks.
Fig. 6.5 and Fig. 6.6 show the coding gain ofM from2to10for the AR(1) process withρ= 0.95 and the AR(2) process withρ= 0.975andθ = π/3, respectively. It is known [107] that forM →
∞the optimal orthonormal SBC approaches the theoretical bound (and therefore all four coders approach the bound asymptotically asM → ∞). However, it can be seen that the performance of the biorthogonal GTD coder is close to the bound even for smallM. It is also interesting to see that this coder class has monotone coding gain behavior with respect to the block sizeM for the AR(1) input. However, such monotone behavior is not present for the AR(2) process. This fact was actually also reported for the traditional orthonormal SBC in the literature [107]. However, It was proved that the coding gain for block sizeM is definitely less than or equal to the coding gain for block sizekM wherekis any positive integer. Fig. 6.5 and Fig. 6.6 suggest that this phenomenon may also be true in the GTD subband coders.
We have shown the usefulness of the frequency dependent GTD in optimizing the perfect re- construction filter banks in subband coders. In the next section we will show that the concept is
2 3 4 5 6 7 8 9 10 6
6.5 7 7.5 8 8.5 9 9.5 10 10.5
M
Coding Gain (dB)
Orthonormal Biorthogonal GTD−Orthonormal GTD−Biorthogonal Theoretical Bound
Figure 6.5: Monotone behavior of the coding gain as a function of the number of channels for the AR(1) process withρ= 0.95.
2 3 4 5 6 7 8 9 10
5 6 7 8 9 10 11 12
M
Coding Gain (dB)
Orthonormal Biorthogonal GTD−Orthonormal GTD−Biorthogonal Theoretical Bound
Figure 6.6: Nonmonotone behavior of the coding gain as a function of the number of channels for the AR(2) process withρ= 0.975andθ=π/3.
applicable in digital communication as well.